The Quadratic Formula (Edexcel IGCSE Maths)

Quadratic Formula

Spec point — 2.7B

What is the quadratic formula?

• A quadratic equation has the form ax² + bx + c = 0 (where a ≠ 0)

  • you need "= 0" on one side

• The quadratic formula is a formula that gives both solutions to a quadratic equation:

$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Exam Hint Make sure the quadratic equation has "= 0" on the right-hand side, otherwise it needs rearranging first.

How do I use the quadratic formula to solve a quadratic equation?

• Read off the values of a, b and c from the equation

• Substitute these into the formula

  • Write this line of working in the exam
  • Put brackets around any negative numbers being substituted in

• To solve 2x² - 8x - 3 = 0 using the quadratic formula:

  • a = 2, b = -8 and c = -3

  • $x = \frac{- ( - 8 ) \pm \sqrt{( - 8 )^{2} - 4 \times 2 \times ( - 3 )} }{2 \times 2}$

  • Either type this into a calculator or simplify by hand

    • Type it once using + for ± then again using - for ±

  • The solutions are x = 4.3452078... or x = -0.34520787....

    • To 3 decimal places: x = 4.345 or x = -0.345
    • To 3 significant figures: x = 4.35 or x = -0.345

Exam Hint Always look for how the question wants you to leave your final answers. For example, correct to 2 decimal places.

How do I write the solutions in an exact (surd) form?

• You may be asked to give answers in an exact (surd) form

• In the example above, work out the number under the square root sign

  • Be careful with negatives!

    • $( - 8 )^{2} - 4 \times 2 \times ( - 3 ) = 64 + 24 = 88$

  • Now square root this number and use surd rules to simplify

    • $\sqrt{88}=\sqrt{4\times22}=\sqrt{4}\times\sqrt{22}=2\sqrt{22}$

  • Substitute this back into the formula and simplify

    • $x = \frac{8 \pm 2 \sqrt{22} }{4} = \frac{2 ( 4 \pm \sqrt{22} )}{4} = \frac{4 \pm \sqrt{22} }{2}$
    • The solutions in exact (surd) form are $x=\frac{4+\sqrt{22}}{2}$ or $x=\frac{4-\sqrt{22}}{2}$

• Calculators that can solve quadratics will give solutions in exact (surd) form

What is the discriminant?

• The part of the formula under the square root (b² – 4ac) is called the discriminant

• The sign of this value tells you if there are 0, 1 or 2 solutions

  • If b²– 4ac > 0 (positive)

    • then there are 2 different solutions

  • If b²– 4ac = 0

    • then there is only 1 solution

  • If b²– 4ac < 0 (negative)

    • then there are no real solutions

• If the discriminant is a perfect square number (0, 1, 4, 9, 16, …) then the quadratic expression can be factorised using integers

Exam Hint You do not need to know about the discriminant for your exam. However, it might be useful to check whether a quadratic can be factorised using integers.

Can I use my calculator to solve quadratic equations?

• If your calculator solves quadratic equations, use it to check your final answers

  • But a correct method and working must still be shown

Worked Example Use the quadratic formula to find the solutions of the equation 3x² - 2x - 4 = 0. Give each solution as an exact value in its simplest form.

Answer:

Write down the values of a, b and c

a = 3, b = -2, c = -4*

Substitute these values into the quadratic formula, $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ Put brackets around any negative numbers

$x = \frac{- ( - 2 ) \pm \sqrt{( - 2 )^{2} - 4 \times 3 \times ( - 4 )} }{2 \times 3}$

Simplify the expressions

$x=\frac{2\pm\sqrt{4+48}}{6}=\frac{2\pm\sqrt{52}}{6}$

Simplify the surd

$x==\frac{2\pm\sqrt{4\times13}}{6}=\frac{2\pm2\sqrt{13}}{6}$

Simplify the fraction

$x=\frac{1\pm\sqrt{13}}{3}$

$x=\frac{1+\sqrt{13}}{3}\mathrm{or}x=\frac{1-\sqrt{13}}{3}$